Area of Surface of Revolution
Most likely one of the last things you studied in your Calculus class was finding the Area of Surface of Revolution. However, it is very important to review this concept and method.
Definition of a Surface of Revolution
If the graph of a continuous function is revolved about a line, the resulting surface is a surface of revolution. Notice the image below.
Definition of the Area of a Surface of Revolution
Let y = f(x) have a continuous derivative on the interval [a, b]. The area S of the surface of revolution formed by revolving the graph of f about a horizontal or vertical axis is
b
S = 2pi § r(x)SQRT(1 + [f '(x)]2) dx
a
where r(x) is the distance between the graph of f and the axis of revolution. If x = g(y) on the interval [c, d], then the surface area is
d
S = 2pi § r(y)SQRT(1 + [g '(y)]2) dy
c
where r(y) is the distance between the graph of g and the axis of revolution.
Click here for an example problem
